arXiv subjects

Gil Cohen

Publications and source records attributed to Gil Cohen.

16 recordsLinked to original sources

arXiv2026

A Forward-Backward Weight Analysis of INW for Permutation Branching Programs

We construct an $\varepsilon$-error PRG for permutation read-once branching programs of length $n$ and width $w$ with seed length \[ O\left((\log w+\log(1/\varepsilon))\cdot \log n\right). \] This gives an exponential improvement in the dependence on $w$ compared with the constructions of De (CCC 2011) and Steinke (ECCC 2012). Compared with the work of Braverman, Rao, Raz, and Yehudayoff (FOCS 2010; SICOMP 2014), which applies more generally to regular branching programs and already achieves the optimal dependence on $w$, our result improves the dependence on the length $n$, attaining the optimal logarithmic dependence. The generator itself is the classical INW PRG of Impagliazzo, Nisan, and Wigderson (STOC 1994). We show that, for permutation branching programs, the INW generator can be instantiated with expanders whose degrees are polynomial in $w$ and $1/\varepsilon$ and, crucially, independent of $n$. To prove this, we analyze error propagation using program-dependent seminorms tailored to the branching program at hand. These seminorms build on the weight function introduced by Braverman et al. The key point is that, when measured in these adapted seminorms, the error does not accumulate throughout the recursion. Since our analysis relies only on the spectral expansion of the underlying expanders, our seed length tightly matches the recent lower bound for spectral analyses of the INW generator due to Hoza, Pyne, and Vadhan (Algorithmica 2024).

cs.CC↗

arXiv2026

Optimal PRGs for Low-Degree Polynomials over Polynomial-Size Fields

Pseudorandom generators (PRGs) for low-degree polynomials are a central object in pseudorandomness, with applications to circuit lower bounds and derandomization. Viola's celebrated construction gives a PRG over the binary field, but with seed length exponential in the degree $d$. This exponential dependence can be avoided over sufficiently large fields. In particular, Dwivedi, Guo, and Volk constructed PRGs with optimal seed length over fields of size exponential in $d$. The latter builds on the framework of Derksen and Viola, who obtained optimal-seed constructions over fields of size polynomial in $d$, although growing with the number of variables $n$. In this work, we construct the first PRG with optimal seed length for degree-$d$ polynomials over fields of polynomial size, specifically $q \approx d^4$, assuming sufficiently large characteristic. Our construction follows the framework of prior work and reduces the required field size by replacing the hitting-set generator used in previous constructions with a new pseudorandom object. We also observe a threshold phenomenon in the field-size dependence. Specifically, we prove that constructing PRGs over fields of sublinear size, for example $q = d^{0.99}$ where $q$ is a power of two, would already yield PRGs for the binary field with comparable seed length via our reduction, provided that the construction imposes no restriction on the characteristic. While a breakdown of existing techniques has been noted before, we prove that this phenomenon is inherent to the problem itself, irrespective of the technique used.

cs.CC↗

arXiv2025

Tracing AG Codes: Toward Meeting the Gilbert-Varshamov Bound

One of the oldest problems in coding theory is to match the Gilbert-Varshamov bound with explicit binary codes. Over larger-yet still constant-sized-fields, algebraic-geometry codes are known to beat the GV bound. In this work, we leverage this phenomenon by taking traces of AG codes. Our hope is that the margin by which AG codes exceed the GV bound will withstand the parameter loss incurred by taking the trace from a constant field extension to the binary field. In contrast to concatenation, the usual alphabet-reduction method, our analysis of trace-of-AG (TAG) codes uses the AG codes' algebraic structure throughout - including in the alphabet-reduction step. Our main technical contribution is a Hasse-Weil-type theorem that is well-suited for the analysis of TAG codes. The classical theorem (and its Grothendieck trace-formula extension) are inadequate in this setting. Although we do not obtain improved constructions, we show that a constant-factor strengthening of our bound would suffice. We also analyze the limitations of TAG codes under our bound and prove that, in the high-distance regime, they are inferior to code concatenation. Our Hasse-Weil-type theorem holds in far greater generality than is needed for analyzing TAG codes. In particular, we derive new estimates for exponential sums.

cs.IT↗

arXiv2025

The Rate-Immediacy Barrier in Explicit Tree Code Constructions

Since the introduction of tree codes by Schulman (STOC 1993), explicit construction of asymptotically good tree codes has remained a notorious challenge. A work by Cohen, Haeupler and Schulman (STOC 2018), as well as the state-of-the-art construction by Ben Yaacov, Cohen, and Yankovitz (STOC 2022) have achieved codes with rate $\Omega(1/\log\log n)$, exponentially improving upon the original rate $\Omega(1/\log n)$ construction of Evans, Klugerman and Schulman from 1994. All of these constructions rely, at least in part, on increasingly sophisticated methods of combining (block) error-correcting codes. In this work, we identify a fundamental barrier to constructing tree codes using known techniques. We introduce a key property which we call immediacy, that, while not required by the original definition of tree codes, is shared by all known constructions and inherently arises in recursive combinations of error-correcting codes. Our main technical contribution is the proof of a rate-immediacy trade-off, which, in particular, implies that any tree code with constant distance and non-trivial immediacy must necessarily have vanishing rate. By applying our rate-immediacy trade-off to existing constructions, we establish that their known rate analyses are essentially optimal given their actual error-correction properties. More broadly, our work highlights the need for fundamentally new ideas -- beyond the recursive use of error-correcting codes -- to achieve substantial progress in explicitly constructing asymptotically good tree codes.

cs.IT↗

arXiv2023

The Ferris ferromagnetic resonance technique: principles and applications

Measurements of ferromagnetic resonance (FMR) are pivotal to modern magnetism and spintronics. Recently, we reported on the Ferris FMR technique, which relies on large-amplitude modulation of the externally applied magnetic field. It was shown to benefit from high sensitivity while being broadband. The Ferris FMR also expanded the resonance linewidth such that the sensitivity to spin currents was enhanced as well. Eventually, the spin Hall angle ({\theta}_SH) was measurable even in wafer-level measurements that require low current densities to reduce the Joule heating. Despite the various advantages, analysis of the Ferris FMR response is limited to numerical modeling where the linewidth depends on multiple factors such as the field modulation profile and the magnetization saturation. Here, we describe in detail the basic principles of operation of the Ferris FMR and discuss its applicability and engineering considerations. We demonstrated these principles in a measurement of the orbital Hall effect taking place in Cu, using an Au layer as the orbital to spin current converter. This illustrates the potential of the Ferris FMR for the future development of spintronics technology.

cond-mat.mes-hall↗

arXiv2020

How moving cracks in brittle solids choose their path

While we fundamentally understand the dynamics of 'simple' cracks propagating in brittle solids within perfect (homogeneous) materials, we do not understand how paths of moving cracks are determined. We experimentally study strongly perturbed cracks that propagate between 10-95\% of their limiting velocity within a brittle material. These cracks are deflected by either interaction with sparsely implanted defects or via an intrinsic oscillatory instability in defect-free media. Dense, high-speed measurements of the strain fields surrounding the crack tips reveal that crack paths are governed by the direction of maximal strain energy density. This fundamentally important result may be utilized to either direct or guide running cracks.

cond-mat.soft↗

arXiv2019

Dynamic fields at the tip of sub-Rayleigh and supershear frictional rupture fronts

The onset of frictional motion at the interface between two distinct bodies in contact is characterized by the propagation of dynamic rupture fronts. We combine friction experiments and numerical simulations to study the properties of these frictional rupture fronts. We extend previous analysis of slow and sub-Rayleigh rupture fronts and show that strain fields and the evolution of real contact area in the tip vicinity of supershear ruptures are well described by analytical fracture-mechanics solutions. Fracture-mechanics theory further allows us to determine long sought-after interface properties, such as local fracture energy and frictional peak strength. Both properties are observed to be roughly independent of rupture speed and mode of propagation. However, our study also reveals discrepancies between measurements and analytical solutions that appear as the rupture speed approaches the longitudinal wave speed. Further comparison with dynamic simulations illustrates that, in the supershear propagation regime, transient and geometrical (finite sample thickness) effects cause smaller near-tip strain amplitudes than expected from the fracture-mechanics theory. By showing good quantitative agreement between experiments, simulations and theory over the entire range of possible rupture speeds, we demonstrate that frictional rupture fronts are classic dynamic cracks despite residual friction.

physics.geo-ph↗

arXiv2017

Topological defects govern crack front motion and facet formation on broken surfaces

Patterns on broken surfaces are well-known from everyday experience, but surprisingly, how and why they form are very much open questions. Well-defined facets are commonly observed1-4 along fracture surfaces which are created by slow tensile cracks. As facets appear in amorphous materials5-7, their formation does not reflect microscopic order. Fracture mechanics, however, predict that slow crack fronts should be straight, creating mirror-like surfaces8-13. In contrast, facet-forming fronts propagate simultaneously within different planes separated by steps. It is therefore unclear why steps are stable, what determines their path and how they couple to crack front dynamics. Here we show, by integrating real-time imaging of propagating crack fronts with surface measurements, that steps are topological defects of crack fronts; crack front separation into discontinuous overlapping segments provides the condition for step stability. Steps drift at a constant angle to the local front propagation direction and the increased local dissipation due to step formation couples to the long-range deformation of the surrounding crack fronts. Slow crack front dynamics are enslaved to changes in step heights and positions. These observations show how 3D topology couples to 2D fracture dynamics to provide a fundamental picture of how patterned surfaces are generated.

cond-mat.soft↗

arXiv2016

Privacy Amplification Against Active Quantum Adversaries

Privacy amplification is the task by which two cooperating parties transform a shared weak secret, about which an eavesdropper may have side information, into a uniformly random string uncorrelated from the eavesdropper. Privacy amplification against passive adversaries, where it is assumed that the communication is over a public but authenticated channel, can be achieved in the presence of classical as well as quantum side information by a single-message protocol based on strong extractors. In 2009 Dodis and Wichs devised a two-message protocol to achieve privacy amplification against active adversaries, where the public communication channel is no longer assumed to be authenticated, through the use of a strengthening of strong extractors called non-malleable extractors which they introduced. Dodis and Wichs only analyzed the case of classical side information. We consider the task of privacy amplification against active adversaries with quantum side information. Our main result is showing that the Dodis-Wichs protocol remains secure in this scenario provided its main building block, the non-malleable extractor, satisfies a notion of quantum-proof non-malleability which we introduce. We show that an adaptation of a recent construction of non-malleable extractors due to Chattopadhyay et al. is quantum proof, thereby providing the first protocol for privacy amplification that is secure against active quantum adversaries. Our protocol is quantitatively comparable to the near-optimal protocols known in the classical setting.

quant-ph↗

arXiv2016

Quantum-Proof Extractors: Optimal up to Constant Factors

We give the first construction of a family of quantum-proof extractors that has optimal seed length dependence $O(\log(n/\varepsilon))$ on the input length $n$ and error $\varepsilon$. Our extractors support any min-entropy $k=\Omega(\log{n} + \log^{1+\alpha}(1/\varepsilon))$ and extract $m=(1-\alpha)k$ bits that are $\varepsilon$-close to uniform, for any desired constant $\alpha > 0$. Previous constructions had a quadratically worse seed length or were restricted to very large input min-entropy or very few output bits. Our result is based on a generic reduction showing that any strong classical condenser is automatically quantum-proof, with comparable parameters. The existence of such a reduction for extractors is a long-standing open question, here we give an affirmative answer for condensers. Once this reduction is established, to obtain our quantum-proof extractors one only needs to consider high entropy sources. We construct quantum-proof extractors with the desired parameters for such sources by extending a classical approach to extractor construction, based on the use of block-sources and sampling, to the quantum setting. Our extractors can be used to obtain improved protocols for device-independent randomness expansion and for privacy amplification.

quant-ph↗

arXiv2015

Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs

In his 1947 paper that inaugurated the probabilistic method, Erd\H{o}s proved the existence of $2\log{n}$-Ramsey graphs on $n$ vertices. Matching Erd\H{o}s' result with a constructive proof is a central problem in combinatorics, that has gained a significant attention in the literature. The state of the art result was obtained in the celebrated paper by Barak, Rao, Shaltiel and Wigderson [Ann. Math'12], who constructed a $2^{2^{(\log\log{n})^{1-\alpha}}}$-Ramsey graph, for some small universal constant $\alpha > 0$. In this work, we significantly improve the result of Barak~\etal and construct $2^{(\log\log{n})^c}$-Ramsey graphs, for some universal constant $c$. In the language of theoretical computer science, our work resolves the problem of explicitly constructing two-source dispersers for polylogarithmic entropy.

math.CO↗

arXiv2015

Crack front dynamics: the interplay of singular geometry and crack instabilities

When fast cracks become unstable to microscopic branching (micro-branching), fracture no longer occurs in an effective 2D medium. We follow in-plane crack front dynamics via real-time measurements in brittle gels as micro-branching unfolds and progresses. We first show that {\em spatially local} energy balance quantitatively describes crack dynamics, even when translational invariance is badly broken. Furthermore, our results explain micro-branch dynamics; why micro-branches form along spatially localized chains and how finite-time formation of cusps along the crack front leads to their death.

cond-mat.mtrl-sci↗

arXiv2014

Two Structural Results for Low Degree Polynomials and Applications

In this paper, two structural results concerning low degree polynomials over finite fields are given. The first states that over any finite field $\mathbb{F}$, for any polynomial $f$ on $n$ variables with degree $d \le \log(n)/10$, there exists a subspace of $\mathbb{F}^n$ with dimension $Ω(d \cdot n^{1/(d-1)})$ on which $f$ is constant. This result is shown to be tight. Stated differently, a degree $d$ polynomial cannot compute an affine disperser for dimension smaller than $Ω(d \cdot n^{1/(d-1)})$. Using a recursive argument, we obtain our second structural result, showing that any degree $d$ polynomial $f$ induces a partition of $F^n$ to affine subspaces of dimension $Ω(n^{1/(d-1)!})$, such that $f$ is constant on each part. We extend both structural results to more than one polynomial. We further prove an analog of the first structural result to sparse polynomials (with no restriction on the degree) and to functions that are close to low degree polynomials. We also consider the algorithmic aspect of the two structural results. Our structural results have various applications, two of which are: * Dvir [CC 2012] introduced the notion of extractors for varieties, and gave explicit constructions of such extractors over large fields. We show that over any finite field, any affine extractor is also an extractor for varieties with related parameters. Our reduction also holds for dispersers, and we conclude that Shaltiel's affine disperser [FOCS 2011] is a disperser for varieties over $F_2$. * Ben-Sasson and Kopparty [SIAM J. C 2012] proved that any degree 3 affine disperser over a prime field is also an affine extractor with related parameters. Using our structural results, and based on the work of Kaufman and Lovett [FOCS 2008] and Haramaty and Shpilka [STOC 2010], we generalize this result to any constant degree.

cs.CC↗

arXiv2013

Bi-Lipschitz Bijection between the Boolean Cube and the Hamming Ball

We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n there exists an explicit bijection f from the n-dimensional Boolean cube to the Hamming ball of equal volume embedded in (n+1)-dimensional Boolean cube, such that for all x and y it holds that distance(x,y) / 5 <= distance(f(x),f(y)) <= 4 distance(x,y) where distance(,) denotes the Hamming distance. In particular, this implies that the Hamming ball is bi-Lipschitz transitive. This result gives a strong negative answer to an open problem of Lovett and Viola [CC 2012], who raised the question in the context of sampling distributions in low-level complexity classes. The conceptual implication is that the problem of proving lower bounds in the context of sampling distributions will require some new ideas beyond the sensitivity-based structural results of Boppana [IPL 97]. We study the mapping f further and show that it (and its inverse) are computable in DLOGTIME-uniform TC0, but not in AC0. Moreover, we prove that f is "approximately local" in the sense that all but the last output bit of f are essentially determined by a single input bit.

math.CO↗

arXiv2006

Crack-Like Processes Governing the Onset of Frictional Slip

We perform real-time measurements of the net contact area between two blocks of like material at the onset of frictional slip. We show that the process of interface detachment, which immediately precedes the inception of frictional sliding, is governed by three different types of detachment fronts. These crack-like detachment fronts differ by both their propagation velocities and by the amount of net contact surface reduction caused by their passage. The most rapid fronts propagate at intersonic velocities but generate a negligible reduction in contact area across the interface. Sub-Rayleigh fronts are crack-like modes which propagate at velocities up to the Rayleigh wave speed, VR, and give rise to an approximate 10% reduction in net contact area. The most efficient contact area reduction (~20%) is precipitated by the passage of slow detachment fronts. These fronts propagate at anomalously slow velocities, which are over an order of magnitude lower than VR yet orders of magnitude higher than other characteristic velocity scales such as either slip or loading velocities. Slow fronts are generated, in conjunction with intersonic fronts, by the sudden arrest of sub-Rayleigh fronts. No overall sliding of the interface occurs until either of the slower two fronts traverses the entire interface, and motion at the leading edge of the interface is initiated. Slip at the trailing edge of the interface accompanies the motion of both the slow and sub-Rayleigh fronts. We might expect these modes to be important in both fault nucleation and earthquake dynamics.

cond-mat.mtrl-sci↗

arXiv1999

Soliton Interaction with an External Traveling Wave

The dynamics of soliton pulses in the Nonlinear Schrodinger Equation (NLSE) driven by an external Traveling wave is studied analytically and numerically. The Hamiltonian structure of the system is used to show that, in the adiabatic approximation for a single soliton, the problem is integrable despite the large number of degrees of freedom. Fixed points of the system are found, and their linear stability is investigated. The fixed points correspond to a Doppler shifted resonance between the external wave and the soliton. The structure and topological changes of the phase space of the soliton parameters as functions of the strength of coupling are investigated. A physical derivation of the driven NLSE is given in the context of optical pulse propagation in asymmetric, twin-core optical fibers. The results can be applied to soliton stabilization and amplification.

patt-sol↗